Geodesic
Multidimensional Euclidean algorithm and continued fractions
Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 7, Tome 538 (2024), pp. 85-101 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

An algorithm is proposed as a multidimensional generalization of the Euclid algorithm. It is similar to the Brun algorithm. For any dimention $d$, the algorithm allows to obtain 1) $d$-dimensional approximations; 2) approximations of linear forms of $d+1$ variables. A verification test of work efficiency was carried out. 
@article{ZNSL_2024_538_a2,
     author = {V. G. Zhuravlev},
     title = {Multidimensional {Euclidean} algorithm and continued fractions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {85--101},
     year = {2024},
     volume = {538},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2024_538_a2/}
}
TY  - JOUR
AU  - V. G. Zhuravlev
TI  - Multidimensional Euclidean algorithm and continued fractions
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2024
SP  - 85
EP  - 101
VL  - 538
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2024_538_a2/
LA  - ru
ID  - ZNSL_2024_538_a2
ER  - 
%0 Journal Article
%A V. G. Zhuravlev
%T Multidimensional Euclidean algorithm and continued fractions
%J Zapiski Nauchnykh Seminarov POMI
%D 2024
%P 85-101
%V 538
%U http://geodesic.mathdoc.fr/item/ZNSL_2024_538_a2/
%G ru
%F ZNSL_2024_538_a2
V. G. Zhuravlev. Multidimensional Euclidean algorithm and continued fractions. Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 7, Tome 538 (2024), pp. 85-101. http://geodesic.mathdoc.fr/item/ZNSL_2024_538_a2/

[1] Khinchin A. Ya., Tsepnye drobi, chetvertoe izd., Nauka, M., 1978

[2] P. Arnoux, S. Labbe, On some symmetric multidimensional continued fraction algorithms, 2015, arXiv: 1508.07814 | MR

[3] V. Berthe, S. Labbe, “Factor complexity of S-adic words generated by the Arnoux–Rauzy–Poincare algorithm”, Advances in Applied Mathematics, 63 (2015), 90–130 | DOI | MR | Zbl

[4] V. Brun, “En generalisation av Kjedebroken”, Skrifter utgit av Videnskapsselskapeti Kristiania, I. Matematisk–Naturvidenskabelig Klasse, 6 (1919–1920)

[5] V. Brun, “Algorithmes euclidiens pour trois et quatre nombres”, Treizieme congres des mathematiciens scandinaves (Helsinki 18–23 aout 1957), Mercators Tryckeri, Helsinki, 1958, 45–64 | MR

[6] J. Cassaigne, “Un algorithme de fractions continues de complexite lineaire”, DynA3S meeting (October 12th, 2015), LIAFA, Paris, 2015

[7] A. Nogueira, “The three-dimensional Poincare continued fraction algorithm”, Israel J. Math., 90:1–3 (1995), 373–401 | DOI | MR | Zbl

[8] E. S. Selmer, “Continued fractions in several dimensions”, Nordisk Nat. Tidskr., 9 (1961), 37–43 | MR | Zbl

[9] F. Schweiger, Multidimensional Continued Fraction, Oxford Univ. Press, New York, 2000 | MR

[10] S. Labbe, $3$-dimensional Continued Fraction Algorithms Cheat Sheets, 2015, arXiv: 1511.08399v1

[11] V. G. Zhuravlev, “Simpleks-yadernyi algoritm razlozheniya v mnogomernye tsepnye drobi”, Sovremennye problemy matematiki, 299, MIAN, 2017, 1–20

[12] V. G. Zhuravlev, Yadernye tsepnye drobi, VlGU, Vladimir, 2019 https://vk.com/id589973164

[13] T. W. Cusick, “Diophantine Approximation of Ternary Linear Forms”, Mathematics of computation, 25:113 (1971), 163–180 | DOI | MR | Zbl

[14] T. W. Cusick, “Diophantine Approximation of Ternary Linear Forms, II”, Mathematics of computation, 26:120 (1972), 977–993 | DOI | MR | Zbl

[15] V. G. Zhuravlev, “Diofantovy priblizheniya lineinykh form”, Algebra i analiz, 490 (2020), 5–24

[16] Dzh. V. S. Kassels, Vvedenie v teoriyu diofantovykh priblizhenii, IL, M., 1961

[17] V. Shmidt, Diofantovy priblizheniya, Mir, M., 1983